$f(x, y) = y^2\ln(\sin(x))$ What is the partial derivative of $f$ with respect to $y$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $y^2 \cot(x)$ (Choice B) B $2y\cot(x)$ (Choice C) C $2y\ln(\sin(x)) + y^2 \cot(x)$ (Choice D) D $2y\ln(\sin(x))$
Answer: Taking a partial derivative with respect to $y$ means treating $x$ like a constant, then taking a normal derivative. $\begin{aligned} \dfrac{\partial f}{\partial y} &= \dfrac{\partial}{\partial y} \left[ {y^2} \ln(\sin(x)) \right] \\ \\ &= {2y} \ln(\sin(x)) \end{aligned}$ In conclusion, $\dfrac{\partial f}{\partial y} = 2y \ln(\sin(x))$